Groschengrab Deluxe 1.31 Cracked |LINK|

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Groschengrab Deluxe 1.31 Cracked

A:

Use “sort” command. It will sort both rows in ascending order of some field. Use’sort’ with -k3, -k2, -k1 and -k0 options for sorting on first, second, third and zero’th field respectively.

Q:

Integral of $\ln(\cosh(\frac x2))$

I have to compute $$\int_{0}^{\infty}\ln(\cosh(\frac x2)) \,dx$$
First, I tried $$\int_{0}^{\infty}\ln(\cosh(\frac x2)) \,dx= \int_{0}^{\infty}\frac{2}{\cosh(x/2)^2} \,dx$$
but after trying to compute the integral, I found that it equals $\infty$.
Then I tried applying the change of variable $u=\frac x2$ to get:
$$2\int_{0}^{\infty}\frac{\cosh(u)}{\cosh(u)} \,du$$
but I could not proceed after this point.
Could someone help me?

A:

You could decompose the integral in two and compute each part:
$$\int_0^{\infty}\ln(\cosh(\frac x2))dx=\int_0^{\frac\pi2}\ln(\cosh(\frac x2))dx+\int_{\frac\pi2}^\infty\ln(\cosh(\frac x2))dx$$
$$=-\frac\pi2\ln(\sinh(\frac \pi2))+\int_{\frac\pi2}^\infty\ln(\cosh(\frac x2))dx$$
$$=-\frac\pi2\ln(\sinh(\frac \pi2))+\int_{0}^{\infty}\ln(\sinh(x))dx$$
$$=-\frac\pi2\ln(\sinh(\frac \pi2))+\ln(2)$$

A:

To complete the complementing answer of PP-expert. I would go for a different integration path, to be precise a deformation of the path of integration. Using the Polar coordinate transformation $x=2\sin u$ (assuming $x>0$) we

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